This is also what used to happen when a citizen of the US (and anyone else who requested it) entered Cuba. Cuban immigration would take a little piece of paper,

This is also what used to happen when a citizen of the US (and anyone else who requested it) entered Cuba. Cuban immigration would take a little piece of paper, place it in the passport, and then stamp the piece of paper. If the traveller happened to remove that piece of paper once the returned from Cuba, so be it.